Tsunami run-ups are associated with massive destructive power and can cause extensive damage to life and property. In 2004, the 9.0 magnitude earthquake that occurred in Sumatra initiated a tsunami which killed over 283 000 people and incurred enormous financial losses (Tanaka, 2009). In 2011, a 9.0 magnitude earthquake occurred in the north of Japan and generated the third-largest tsunami in the past ten years; the tsunami submerged 400 square kilometres of land and killed over 20 000 people (Mori et al., 2011). Hence, it is necessary to determine effective methods to decrease tsunami run-up. A traditional and direct method to protect coastal regions from wave run-up is the construction of hard structures, such as seawalls and break-waters; however, this approach is expensive and harmful to the environment. Therefore, research regarding the mitigation of damage due to tsunami run-ups by using structures with low environmental impact has become quite prevalent. Coastal vegetation can realize reasonable wave run-up dissipation with less environmental disruption than that associated with artificial engineering (Zhang et al., 2010, 2013; Duarte et al., 2013; Temmerman et al., 2013). However, in nature, coastal vegetation is rarely distributed uniformly and continuously; instead, it tends to be distributed in patches owing to regrowth, decay and man-made damage. Tsunamis are always simulated by a solitary wave; thus, it is necessary to investigate the attenuation of a solitary wave run-up by patchy vegetation to analyse the influence of patchy vegetation on the tsunami run-up.

In recent years, several researchers have investigated the effect of vegetation on wave run-up attenuation. Irtem et al. (2009) found that wave run-up was reduced by approximately 45% on a vegetated slope compared to that on a landscape without trees. Tang et al. (2013) investigated the sensitivity of solitary wave run-up to plant height, diameter and stem density by comparing the numerical results pertaining to different patterns of vegetation. Tang et al. (2017) studied the effects of vegetation on long-period wave run-up via a numerical simulation by using the nonlinear shallow water equations, and the results indicated that the attenuation of long-period wave run-up due to vegetation is sensitive to the variation of the incident wave period; in addition, the attenuation of wave run-up did not increase or decrease monotonically with the incident wave period. Tsai et al. (2017) studied the damping effect of solitary wave propagation on moving emergent vegetation by using a numerical model based on the general Reynolds-averaged Navier-Stokes equations; the results indicated that the effect of moving vegetation on the damping of the wave height was notably less than the corresponding effect of stationary vegetation. Yao et al. (2018), on the basis of a modified Boussinesq equation, investigated the effects of beach slope, forest density and tree distribution on the solitary wave drag coefficient. Thuy et al. (2018) proposed a numerical model based on 1D nonlinear longwave equations, and the model was applied to investigate the effects of forest width, tree density, incident tsunami period and height on the height of wave run-up. The existing have focused mainly on the effect of continuous and uniform vegetation on solitary run-up. In nature, coastal vegetation is always distributed in patches due to regrowth, decay or man-made damage, and its internal distribution is often non-uniform (Maza et al., 2016); however, only a few studies have focused on patchy vegetation. Fonseca and Koehl (2006) and Vandenbruwaene et al. (2011) studied the effects of patch sizes and inter-patch distance on currents. Irish et al. (2014) evaluated the reduction in the intensity of a tsunami via patchy vegetation by conducting a series of experiments. Maza et al. (2016) proposed a new parameter termed the equivalent length, which can take the effect of patchiness of vegetation into analysis. Using this parameter, the authors investigated the effect of the submergence ratio and the field characteristic of circular vegetation patches on wave attenuation. Yang et al. (2017) investigated the behaviour of a tsunami when coming ashore through patchy vegetation and the variation of this behaviour with the roughness level, patch spacing and patch size. Zainali et al. (2018) performed numerical simulations of long waves interacting with arrays of emergent cylinders inside regularly spaced patches. The abovementioned studies revealed the effects of uniform or patched vegetation on the attenuation of solitary wave run-up. However, the effects of density and internal distribution of vegetation patches on solitary wave run-up remain yet unclarified.

In this study, the effect of patchy vegetation on the attenuation of solitary wave run-up was investigated by using a wave-vegetation numerical model based on COULWAVE (which is based on nonlinear Boussinesq equations). The paper is organized as follows. Section 2 presents the governing equations for wave propagation in vegetation zones, which are based on the Boussinesq model. Section 3 describes the analysis of the accuracy of the numerical model by comparing the numerical results with the laboratory data pertaining to wave propagation with continuous vegetation on a flat bottom surface and patched vegetation on a slope; this section also describes the investigation pertaining to the sensitivity of solitary wave run-up to the distribution of patchy vegetation. Section 4 presents the conclusions derived from this study and the limitations.

Boussinesq equations have been widely used for modelling coastal waves (Liu and Fang, 2016, 2019; Liu et al., 2018; Yang et al., 2018). The governing equations of COULWAVE are high-order nonlinear Boussinesq equations. In this study, the governing equations based on COULWAVE, including the vegetation term can be described as follows (Lynett et al., 2002; Kim et al., 2009):

where ζ is the wave elevation; H=ζ+h is the total water depth, where h is the still water depth; g is the gravity acceleration; U_α and V_α respectively represent the horizontal velocity in the x and y directions at any horizontal plane; D_x and D_y respectively refer to the second-order term of the horizontal momentum equation in the x and y directions of average depth; and f_veg is the force acting on the vegetation.

In this study, the plants are represented by cylinders, and the Morison equation is adopted to indicate the force acting on the vegetation. The inertial term is ignored, as the diameter of vegetation is considerably small compared to the wavelength. The force acting on vegetation can be expressed as

where C_d is the drag coefficient, N_v is the vegetation density, b_v is the vegetation diameter, and ${{\mathop u\limits^ \rightharpoonup} _p}$ is the horizontal velocity in the vegetation region. The relationship between ${{\mathop u\limits^ \rightharpoonup} _p}$ and ${{\mathop u\limits^ \rightharpoonup}}$ can be expressed as ${{\mathop u\limits^ \rightharpoonup} _p} = {\mathop u\limits^ \rightharpoonup} /(1 - \phi)$, where $\phi$ is the vegetation volume fraction, $\phi = {V_s}/V$, V_s is the vegetation volume and V is the control volume.

In the governing equation, the vegetation drag coefficient is the most important parameter for simulating the vegetation drag force, and its magnitude depends on the characteristics of the fluid and vegetation. In this study, the equation derived by Kobayashi et al. (1997) is employed:

where R_d=|u_c|b_v/γ, F=|u_c|/(gd)^1/2, γ=1.01×10^-6 m²/s is the hydrodynamic viscosity, ξ=0.8, u_c is the maximal velocity in the vegetation region (Stone and Shen, 2002), R_d is the Reynolds number and F is the Froude number.

When simulating wave propagation, the existence of offshore structures and the change in submarine topographies can influence the process of wave propagation, and some of these aspects can even cause wave breaking. Hence, it is vital to simulate the bottom fraction, wave breaking and boundary conditions accurately. The wave breaking term and bottom fraction term were introduced in the momentum equation:

where R_f is the bottom fraction term, and R_b is the wave breaking term.

For the bottom fraction term, R_f can be expressed as:

where f is the bottom fraction coefficient, H is the total water depth and ${{\mathop u\limits^ \rightharpoonup} _b}$ is the bottom horizontal water velocity. The bottom fraction coefficient f is related to Chezy’s coefficient, f=g/C²; when the seabed is rough, C<10, and when the seabed is typical, 20<C<60.

For the wave breaking term, R_b can be expressed as:

where

where υ is the motion viscosity coefficient, and υ=BHζ_t. The value of variable B can be determined as follows:

where δ is an amplification factor, and ζ_t is defined as:

where ζ_t^(I) is the initial displacement when wave breaking occurs, ζ_t^(F) is the minimum displacement to maintain wave breaking, t₀ is the time when wave breaking is initiated, and T^b is the instantaneous time. In terms of contrast with the laboratory data (Hansen and Svendsen, 1979), the parameters above can be verified as:

First, the accuracy of the model is validated by simulating a solitary wave run-up on patchy vegetation on flat bottom and continuous vegetation on slope. Second, the model is used to study the effect of patchy vegetation on wave run-up on a slope by performing a simulation of wave propagation in the presence of different internal distributions.

In this study, the effects of patchy vegetation on solitary wave run-up were investigated by using a numerical simulation. The numerical model is based on an implementation of Morison's formulation for rigid-structure-induced drag stresses in the COULWAVE model. The accuracy of the model was examined by conducting laboratory experiments pertaining to solitary wave propagation through patchy vegetation on a flat bottom and continuous vegetation on a slope. Subsequently, the model was applied to an investigation of the solitary wave run-up through a patchy vegetation slope. The simulation results indicated that patchy vegetation could lead to a considerable reduction in the wave run-up. Patchy vegetation can reduce the wave run-up to approximately 60% when compared to condition without vegetation. Furthermore, it was noted that the internal distribution can influence the attenuation effect of the solitary wave run-up, and high-density patched vegetation can attenuate the solitary wave run-up more effectively than low-density patched vegetation can. For vegetation patches with the same density, patched vegetation with uniform distribution has a better effect on the attenuation wave run-up when compared to the effect of a non-uniform distribution.

It should be noted that in the present study, the vegetation is represented by rigid sticks, which is a simplification compared to real vegetation. Natural vegetation is more complex due to the presence of plant branches and leaves, the obvious non-rigidity of the plants, and the spatial or seasonal variations in plant properties. In future work, laboratory data using real vegetation or field data may be used to further validate the proposed model and to confirm the validity of our findings under naturally vegetated coastal environments.